Question

Tamara, Payton, Krish, and Haruki are analyzing the solution sets to the following linear equations. Tamara says that Equation A and Equation C have the same solution. Payton says Equation B and Equation C have the same solution. Krish says all three have the same solution. Haruki says that none of the equations have the same solution. Identify who is correct about the linear equations.

Equation A: 3(4−5h)−7=−12+2h

Equation B: 2(p−3)−1= 6p+14/2

Equation C: 2/3 v−3+ 8/3 =−1

(1 point)
Responses

Payton
Payton

Krish
Krish

Haruki
Haruki

Tamara
Tamara

Answer 1

To determine who is correct about the solution sets of the equations, we need to solve each equation step by step.

**Equation A:**
\[ 3(4 - 5h) - 7 = -12 + 2h \]
Expanding and simplifying:
\[ 12 - 15h - 7 = -12 + 2h \]
\[ 5 - 15h = -12 + 2h \]
Adding \(15h\) to both sides:
\[ 5 = -12 + 17h \]
Adding \(12\) to both sides:
\[ 17 = 17h \]
Dividing by \(17\):
\[ h = 1 \]

**Equation B:**
\[ 2(p - 3) - 1 = \frac{6p + 14}{2} \]
Expanding the left side:
\[ 2p - 6 - 1 = \frac{6p + 14}{2} \]
Simplifying further:
\[ 2p - 7 = \frac{6p + 14}{2} \]
To eliminate the fraction, multiply both sides by \(2\):
\[ 2(2p - 7) = 6p + 14 \]
This gives:
\[ 4p - 14 = 6p + 14 \]
Rearranging gives:
\[ -14 - 14 = 6p - 4p \]
\[ -28 = 2p \]
Dividing by \(2\):
\[ p = -14 \]

**Equation C:**
\[ \frac{2}{3}v - 3 + \frac{8}{3} = -1 \]
Combining terms on the left:
\[ \frac{2}{3}v + \frac{8}{3} - 3 = -1 \]
Convert \(3\) into thirds:
\[ \frac{2}{3}v + \frac{8}{3} - \frac{9}{3} = -1 \]
This simplifies to:
\[ \frac{2}{3}v - \frac{1}{3} = -1 \]
Adding \(\frac{1}{3}\) to both sides:
\[ \frac{2}{3}v = -1 + \frac{1}{3} \]
Converting \(-1\) into thirds:
\[ \frac{2}{3}v = -\frac{3}{3} + \frac{1}{3} \]
This gives:
\[ \frac{2}{3}v = -\frac{2}{3} \]
Multiplying both sides by \(\frac{3}{2}\):
\[ v = -1 \]

Now we have the following solutions:
- Equation A \(h = 1\)
- Equation B \(p = -14\)
- Equation C \(v = -1\)

Based on the solutions:
- Tamara's claim that Equation A and Equation C have the same solution is **incorrect**.
- Payton's claim that Equation B and Equation C have the same solution is **incorrect**.
- Krish's claim that all three have the same solution is **incorrect**.
- Haruki's claim that none of the equations have the same solution is **correct**.

Therefore, the correct answer is **Haruki**.